How to complete this hole with the grid around it?
First start by making the hole match up with the underlying grid. In my case I Filled the two faces on each side, then deleted the one in the corner.



Now it is a simple matter of using the Bridge Edge Loops operator. Select the vertices along the two long edges, but not the corner vertices. Like this:


Run Bridge Edge Loops, either from the 3D view header Mesh > Edges > Bridge Edge Loops, or from the specials menu W Bridge Edge Loops. Wait! Before doing anything else open up the operator history (bottom part of the tool shelf or press F6). Change the Number of Cuts to match the topology (in my case it was three), and set the Interpolation to Blend Surface.


Ah that's looking better.

Last thing to do is fill the remaining holes. That is easily done in one step. Just select the edges (like I show in the image below) and run Bridge Edge Loops again.